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Enumeration of Non-Orientable 3-Manifolds Using Face Pairing Graphs and Union-Find∗

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Enumeration of Non-Orientable 3-Manifolds Using Face Pairing Graphs and Union-Find∗ Enumeration of non-orientable 3-manifolds using face pairing graphs and union-find∗ Benjamin A. Burton Author’s self-archived version Available from http://www.maths.uq.edu.au/~bab/papers/ Abstract Drawing together techniques from combinatorics and computer science, we improve the census algorithm for enumerating closed minimal P2-irreducible 3-manifold triangulations. In particular, new constraints are proven for face pairing graphs, and pruning techniques are improved using a modification of the union-find algorithm. Using these results we catalogue all 136 closed non-orientable P2-irreducible 3-manifolds that can be formed from at most ten tetrahedra. 1 Introduction With recent advances in computing power, topologists have been able to construct exhaustive tables of small 3-manifold triangulations, much like knot theorists have constructed exhaustive tables of simple knot projections. Such tables are valuable sources of data, but they suffer from the fact that enormous amounts of computer time are required to generate them. Where knot tables are often limited by bounding the number of crossings in a knot pro- jection, tables of 3-manifolds generally limit the number of tetrahedra used in a 3-manifold triangulation. A typical table (or census) of 3-manifolds lists all 3-manifolds of a particular type that can be formed from n tetrahedra or fewer. Beyond their generic role as a rich source of examples, tables of this form have a number of specific uses. They still offer the only general means for proving that a triangulation is minimal (i.e., uses as few tetrahedra as possible), much in the same way as knot tables are used to calculate crossing number. Moreover, a detailed analysis of these tables can offer insight into the combinatorial structures of minimal triangulations, as seen for example by the structural observations of Matveev [20], Martelli and Petronio [17] and Burton [8]. Unfortunately the scope of such tables is limited by the difficulty of generating them. In general, a census of triangulations formed from ≤ n tetrahedra requires computing time at least exponential in n. In the case of closed 3-manifold triangulations, results are only known for ≤ 11 tetrahedra in the orientable case and ≤ 8 tetrahedra in the non-orientable case. These results are particularly sparse in the non-orientable case — only 18 distinct manifolds are found, all of which are graph manifolds [7]. Clearly there is more to be learned by extending the existing censuses to higher numbers of tetrahedra. Due to the heavy computational requirements however, this requires significant improvements in the algorithms used to generate the census data. Such improvements form the main subject of this paper. We restrict our attention here to closed 3-manifold triangulations. In the orientable case, successive tables have been generated by Matveev [20], Ovchinnikov, Martelli and Petronio [15], Martelli [14] and then Matveev again with 11 tetrahedra [21]. For non-orientable manifolds, ∗This project was supported by the Victorian Partnership for Advanced Computing e-Research Program Grant Scheme. 1 tabulation begins with Amendola and Martelli [2, 3] and is continued by Burton [8, 7] up to 8 tetrahedra. The contributions of this paper are the following: • Several improvements to the algorithm for generating census data, some of which increase the speed by orders of magnitude; • An extension of the closed non-orientable census from 8 tetrahedra to 10 tetrahedra; • A verification of previous closed orientable census results for up to 10 tetrahedra, and an extension of these results from a census of manifolds to a census of all minimal triangulations. The algorithmic improvements are divided into two broad categories. The first set of results relate to face pairing graphs, which are 4-valent graphs describing which tetrahedron faces are identified to which within a triangulation. The second set is based upon the union- find algorithm, which is a well-known method for finding connected components in a graph. Here the union-find algorithm is modified to support backtracking, and to efficiently monitor properties of vertex and edge links within a triangulation. All computational work was performed using the topological software package Regina [4, 6]. Census generation forms only a small part of Regina, which is a larger software package for performing a variety of tasks in 3-manifold topology. The software is released under the GNU General Public License, and may be freely downloaded from http://regina.sourceforge. net/. The bulk of this paper is devoted to improving the census algorithm. Section 2 begins with the precise census constraints, and follows with an overview of how a census algorithm is structured. In Section 3 we present a series of preliminary results, describing properties of minimal triangulations that will be required in later sections. Section 4 offers the first round of algorithmic improvements, based upon the analysis of face pairing graphs. A more striking set of improvements is made in Section 5, in which a modified union-find algorithm is used to greatly reduce the search space. Both Sections 4 and 5 also include empirical results in which the effectiveness of these improvements is measured. The improvements of Sections 4 and 5 have led to new closed census results, as outlined above. Section 6 summarises these new results, with a focus on the extension of the closed non-orientable census from 8 to 10 tetrahedra. A full list of non-orientable census manifolds is included in the appendix. 2 Overview of the Census Algorithm As is usual for a census of closed 3-manifolds, we restrict our attention to manifolds with the following properties: • Closed: The 3-manifold is compact, with no boundary and no cusps. • P2-irreducible: The 3-manifold contains no embedded two-sided projective planes, and every embedded 2-sphere bounds a ball. The additional constraint of P2-irreducibility allows us to focus on the most “fundamental” manifolds — the properties of larger manifolds are often well understood in terms of their P2- irreducible constituents. Recall that we are not just enumerating 3-manifolds, but also their triangulations. Through- out this paper we consider a triangulation to be a finite collection of n tetrahedra, where some or all of the 4n tetrahedron faces are affinely identified in pairs. For the census we focus only on triangulations with the following additional property: • Minimal: The triangulation uses as few tetrahedra as possible. That is, the underlying 3-manifold cannot be triangulated using a smaller number of tetrahedra. This minimality constraint is natural for a census, and is used throughout the literature. Note that a 3-manifold may have many different minimal triangulations, though of course all of these triangulations must use the same number of tetrahedra. Minimal triangulations are tightly related to the Matveev complexity of a manifold [19]. Matveev defines complexity in terms of special spines, and it has been proven by Matveev in 2 the orientable case and Martelli and Petronio in the non-orientable case [16] that, with the 3 3 2 exceptions of S , RP and L3,1, the Matveev complexity of a closed P -irreducible 3-manifold is precisely the number of tetrahedra in its minimal triangulation(s). 2.1 Stages of the Algorithm There are two stages involved in constructing a census of 3-manifold triangulations: the gen- eration of triangulations, and then the analysis of these triangulations. 1. Generation: The generation stage typically involves a long computer search, in which tetrahedra are pieced together in all possible ways to form 3-manifold triangulations that might satisfy our census constraints. The result of this search is a large set of triangulations, guaranteed to include all of the triangulations that should be in the census. There are often unwanted triangulations also (for instance, triangulations that are non-minimal, or that represent reducible manifolds). This is not a problem; these unwanted triangulations will be discarded in the analysis stage. The generation of triangulations is entirely automated, but it is also extremely time- consuming — it may take seconds or centuries, depending upon the size of the census. 2. Analysis: Once the generation stage has produced a raw set of triangulations, these must be refined into a final census. This includes verifying that each triangulation is minimal and P2-irreducible (and throwing away those triangulations that are not). It also involves grouping triangulations into classes that represent the same 3-manifold, and identifying these 3-manifolds. Analysis is much faster than generation, but it typically requires a mixture of automa- tion and human involvement. Techniques include the analysis of invariants and normal surfaces, combinatorial analysis of the triangulation structures, and applying elementary moves that change triangulations without altering their underlying 3-manifolds. The generation stage is the critical bottleneck, due to the vast number of potential trian- gulations that can be formed from a small number of tetrahedra. Suppose we are searching for triangulations that can be formed using n tetrahedra. Even assuming that we know which tetrahedron faces are to be joined with which, each pair of faces can be identified according to one of six possible rotations or reflections, giving rise to 62n possible triangulations in total. For 10 tetrahedra, this figure is larger than 1015. It is clear then why existing census data is limited to the small bounds that have been reached to date. It should be noted that for an orientable census, the figure 62n becomes closer to 3n6n. This is because for a little over half the faces only three of the six rotations or reflections will preserve orientation. It is partly for this reason that the orientable census has consistently been further advanced in the literature than the non-orientable census. Nevertheless, for 10 tetrahedra this figure is still larger than 1012, a hefty workload indeed.
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